System and method for image processing with highly undersampled imaging data

ABSTRACT

A system and method for processing highly undersampled multi-echo spin-echo data by linearizing the slice-resolved extended phase graph model generates highly accurate T2 maps with indirect echo compensation. Principal components are used to linearize the signal model to estimate the T2 decay curves which can be fitted to the slice-resolved model for T2 estimation. In another example of image processing for highly undersampled data, a joint bi-exponential fitting process can compensate for image variations within a voxel and thus provide partial voxel compensation to produce more accurate T2 maps.

FEDERAL FUNDING ACKNOWLEDGEMENT

This invention was made with government support under Grant/Contract No.R01 HL085385 awarded by the National Institutes of Health. Thegovernment has certain rights in the invention.

BACKGROUND OF THE INVENTION

Field of the Invention

The present disclosure relates generally to image processing and, morespecifically, to a system and method for image processing using highlyundersampled imaging data.

Description of the Related Art

Spin-spin (T₂) relaxation is one of the main contrast mechanisms in MRI.Although most clinical applications use qualitative (visual) informationderived from T₂-weighted images, there is an increasing interest in T₂mapping (1-10).

Because single-echo spin-echo T₂ mapping requires long acquisitiontimes, its translation to the clinic has been limited by its timeinefficiency. In order to reduce the acquisition times it is customaryto use multi-echo spin-echo (MESE) pulse sequences, where several echotime (TE) points are acquired per repetition time (TR) period by using atrain of 180° refocusing pulses after the initial 90° excitation pulse.To further accelerate T₂ data acquisition, a fast (or turbo) spin-echoapproach where several k-space lines of data are acquired per TR periodis commonly used. For the sake of speed in T₂ mapping (while maintaininghigh spatial and temporal resolution), the use of TE data sets that areundersampled in k-space has been proposed in conjunction with a fastspin-echo approach. Several algorithms have been described to recover T₂information from these highly reduced TE data sets (11-16). Recently,the focus has been on model-based T₂ mapping algorithms. Doneva et al.proposed to exploit the temporal sparsity of the exponential decay whilereconstructing all TE images under the framework of compressed sensing(13, 16). Block et al. proposed a model-based algorithm for radialfast-spin-echo acquisitions to directly reconstruct I₀ and R₂ (1/T₂)maps from the measured k-space data (11). Because I₀ and R₂ values havevery different scales the gradient-based minimization process requires ascaling factor which needs to be fine-tuned for accurate T₂ estimation(12). Our group has recently developed the REPCOM (REconstruction ofPrincipal COmponent coefficient Maps) algorithm which linearizes thesignal model using principal component analysis (PCA). REPCOM exploitsthe spatial and temporal sparsity of the TE images, and providesaccurate T₂ estimates from highly undersampled data without the need ofa scaling factor for the fitted parameters.

The algorithms described above, including REPCOM, rely on the assumptionthat the signal follows an exponential decay. However, in MESEacquisitions the decay is generally contaminated by indirect echoes(echoes leading to signal generation after more than one refocusingpulse such as stimulated echoes) including differences in the signalintensities between even and odd echoes, thus, altering the singleexponential nature of the T₂ decay observed in a single-echo spin-echoexperiment. The indirect echoes are the result of refocusing pulses notattaining the ideal 180° flip angle (FA) due to nonrectangular sliceprofiles, static (B₀) and transmit field (B₁) inhomogeneity, and B₁calibration errors (17).

To alleviate the pulse imperfection due to nonrectangular sliceprofiles, a thick refocusing slice technique has been proposed by Pellet al. (5). This technique employs a refocusing slice that is thickerthan the excitation slice. B₀ and B₁ inhomogeneity, and calibrationerrors, however, are not corrected for by this approach. Echo editingtechniques that use crusher gradients around the refocusing pulses havealso been proposed to reduce the signal resulting from pathways leadingto indirect echoes (18-20). However, not all pathways can be crushedeffectively and the method has only been demonstrated with non-selectiverefocusing pulses, which limits the use of the method to single sliceapplications. Recently, Lebel and Wilman proposed the slice-resolvedextended phase graph (SEPG) fitting algorithm (17), for accurate T₂estimation from MESE data contaminated by indirect echoes. Their methodis based on the extended phase graph (EPG) model proposed by Hennig (21)which provides decay curves for any given refocusing FA. The EPG modelassumes perfectly rectangular slice profiles whereas the SEPG modelincludes the known slice profile for both excitation and refocusingpulses. The fitting algorithm fits the measurements to the SEPG model toobtain the T₂ estimates. The method is robust to B₁ inhomogeneity andcalibration errors and it has been shown that accurate T₂ estimation canbe obtained from MESE data acquired with reduced FAs (<1800).

So far, the SEPG fitting algorithm has been demonstrated for fullysampled or 60% partial k-space Cartesian data. The main limitation ofcombining the SEPG fitting algorithm with a model-based reconstructionapproach for T₂ estimation from highly undersampled data (<10% sampledwith respect to Nyquist sampling theorem), is the non-linearity of theSEPG model. Therefore it can be appreciated that there is a significantneed for a technique to provide proper fitting with highly undersampleddata. The present disclosure provides this and other advantages as willbe apparent from the following detailed description and accompanyingfigures.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWING(S)

FIG. 1 is a block diagram of an exemplary system used to implement thetechniques of the present disclosure.

FIG. 2 is a flow chart illustrating the signal processing by the systemof FIG. 1.

FIGS. 3A-3C are a series of graphs illustrating selected trainingcurves, the approximation error for the training curves, and thedistribution of the percentage of error caused by the approximationerror.

FIGS. 4A-4D illustrate the decay curves for phantom tube sample dataacquired with different re-focusing flip angles (FA) and standardsampling (64% relative to the Nyquist condition).

FIG. 5 is a data table illustrating the percent error of T₂ estimatesfor phantom data acquired with standard sampling for various FA.

FIGS. 6A-6B illustrate the errors in the decay curves of the threephantom tubes illustrated in FIG. 4 but acquired with a high degree ofundersampling (4% relative to the Nyquist condition).

FIG. 7 illustrates a data table showing the percent error of T₂estimates for highly undersampled data that correspond to the samephysical phantom used in FIGS. 4-6 where the percent errors are relativeto the gold standard.

FIG. 8 illustrates an MRI image with a designated region of interest(ROI) and the decay curves and T₂ estimates of the ROI for different FAvalues.

FIG. 9 shows an image of a T₂ map obtained using the present techniquesand the % difference between T2 maps obtained from data acquired withprescribed FAs of 180 and 120 using REPCOM (a technique which does notcompensate for indirect echoes) and CURLIE with SEPG fitting with twodifferent T1 values.

FIG. 10 is a series of abdominal images with a T₂ map overlay onto ananatomical image with the T₂ maps being generated by REPCOM as well asthe techniques described herein.

FIG. 11 is a data table illustrating the percent error of T₂ estimatesfor the phantoms acquired with the liver data illustrated in FIG. 10 inwhich the T₂ curves were reconstructed from undersampled data (4%sampled with respect to the Nyquist criteria) using REPCOM and theprocessing of the present disclosure.

FIG. 12 is a series of cardiac images with anatomical images and anoverlay of the T₂ map using the present techniques and REPCOM.

FIG. 13 is a table illustrating the bar plots of estimated data versuslesion diameter for various fitting algorithms.

FIGS. 14A-14C are a series of graphs illustrating T₂ errors estimated byvarious fitting algorithms for a tight region of interest and anexpanded region of interest.

FIG. 15 is a data table illustrating the results from a physical phantomshowing the percent error in T₂ estimates for various fitting algorithmsfor both a tight and expanded region of interest.

FIG. 16 is a data table illustrating the results from a physical phantomshowing T₂ estimates for the fitting processing described herein usinghighly undersampled data.

FIG. 17 is an image illustrating a lesion classification plot generatedusing data from a set of subjects with liver lesions where the T_(2L)values were estimated by the fitting algorithm described herein and asingle exponential fitting from highly undersampled data acquired in onebreath hold.

DETAILED DESCRIPTION OF THE INVENTION

The present disclosure is directed to a novel reconstruction techniquefor the estimation of the T₂, or spin-spin relaxation time, using highlyundersampled MRI data acquired with a fast spin-echo (FSE) pulsesequence. The FSE sequence has the advantage that data acquisition isfast and that the temporal data sets used for calculating the T₂ mapsare registered by the nature of the acquisition. As used herein, highlyundersampled data refers to a sampling rate that is significantly lessthan the conventional Nyquist sampling rate criteria. For example,highly undersampled data may be in the range of 4-10% sample ratecompared to the Nyquist sampling criteria. The highly undersampled datapermits faster T₂ mapping; the use of a radial k-space trajectory allowsfor greater flexibility in the reconstruction of T₂ maps. A drawback ofthe FSE acquisition is the presence of indirect echoes which introduceerrors in T₂ estimation when the signal is modeled by an exponentialcurve. The SEPG model, proposed by Lebel et al, is a betterrepresentation of the T₂ signal decay in the presence of indirectechoes. (17) However, the high non-linearity of the SEPG model impairsthe reconstruction of T₂ maps when highly undersampled data are used.

The techniques described herein use a principal-component analysis (PCA)approach to linearize the SEPG signal and use an iterativereconstruction approach which incorporates the principles of compressedsensing (CS) to recover T₂ maps from highly undersampled data withindirect echo compensation. The results are accurate T₂ maps obtainedfrom data with high temporal and spatial resolution from data acquiredis a short period of time (e.g., breath hold). This is particularlyimportant for T₂ mapping in areas in the body affected by motion, suchas the abdomen and thoracic cavity.

The idea of using a linear approximation to the signal model togetherwith CS to reconstruct parameter maps from highly undersampled data canbe extended beyond the SEPG model and the FSE acquisition sequence. Ingeneral, signal models could be based on the Bloch equations.

While the example presented herein utilizes principal components as alinear approximation however, this is just one possibility. Otherapproximations can be used such as dictionaries, manifolds, and thelike.

Those skilled in the art will appreciate that the linearized approachdoes not directly yield a parameter map, but coefficient maps from whichthe parameter can be estimated. This provides significant flexibilityfor the fitting of the data. For example, a least squares fittingapproach can be used to obtain the parameter maps. More complex methodssuch as algorithms that deal with partial volume effects (caused bymultiple components within a voxel) can also be used. In one embodiment,the fitting of parameters in multi-component systems can besignificantly improved by using a joint estimation algorithm describedherein. The joint estimation processing described below was developedwith the goal of estimating parameter maps for small lesions embedded inan organ where partial volume effects are more pronounced.Characterizing small lesions is particularly important for earlydetection of cancer.

In one embodiment, it is possible to extend the PCA approach used in theREPCOM algorithm to linearize the SEPG model. The CUrve Reconstructionvia pca-based Linearization with Indirect Echo compensation (CURLIE)technique described herein aims to obtain accurate T₂ decay curves fromhighly undersampled data in the presence of B₁ imperfections or non-180°refocusing pulses. The T₂ values of the reconstructed curves can then beobtained by applying SEPG fitting. Although only radial data is used inthis work, the methodology can be translated to other trajectories.

Hennig's EPG model provides a way to calculate the signal intensity of aspecific echo point for MESE sequences with the assumption that allspins experience the same FA. Given the excitation pulse FA, α₀, the FAsof the N refocusing pulses, α_(j) (j=1, . . . , N), the sensitivity oftransmit B₁ field, and the I₀, T₁ and T₂ values of a voxel, the signalintensity at the j^(th) spin echo, S_(j), can be expressed as accordingto ref. 21:S _(j) =I ₀ ·EPG(T ₁ ,T ₂ ,B ₁,α₀, . . . ,α_(j) ,j).  [1]The function EPG(●) does not have an explicit expression, but can becomputed numerically. Based on the EPG model, Lebel and Wilman recentlyproposed the SEPG model (17). With the prior knowledge of the prescribedslice profile of the excitation pulse along the z direction, α₀(z), theprescribed slice profiles of the refocusing pulse along the z direction,α_(j)(z) (j=1, . . . , N), the SEPG model can be obtained by integratingthe EPG model throughout the slice:

$\begin{matrix}{{{C_{j}\left( {I_{0},T_{1},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)}} \right)} = {I_{0} \cdot {\int_{z}{{{EPG}\left( {T_{1},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)},j} \right)}{{dz}.}}}}}\;} & \lbrack 2\rbrack\end{matrix}$The SEPG fitting algorithm fits T₂ decay curves acquired with a MESEsequence to Equation [2]. Given NTE measurements s_(j) (j=1, . . . , N),the SEPG fitting algorithm can be written as:

$\begin{matrix}{I_{0},T_{1},T_{2},{B_{1} = {\arg\;{\min_{I_{0},T_{1},T_{2},B_{1}}{\left\{ {\sum\limits_{j = 1}^{N}\;{{{C_{j}\left( {I_{0},T_{1},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)}} \right)} - s_{j}}}^{2}} \right\}.}}}}} & \lbrack 3\rbrack\end{matrix}$In Ref. 17, it has been shown that the SEPG model is insensitive to T₁values when T₁/T₂ ratio is large, hence it has been proposed to fix T₁to +∞ in order to simplify the signal model and the fitting algorithm:

$\begin{matrix}{{{C^{\prime}\left( {I_{0},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)},j} \right)} = {I_{0} \cdot {\int_{z}{{{EPG}\left( {{T_{1} = {+ \infty}},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)},j} \right)}{\mathbb{d}z}}}}},} & \lbrack 4\rbrack \\{I_{0},T_{2},{B_{1} = {\arg\;{\min_{I_{0},T_{2},B_{1}}{\left\{ {\sum\limits_{j = 1}^{N}\;{{{C_{j}^{\prime}\left( {I_{0},T_{1},{= {+ \infty}},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)}} \right)} - s_{j}}}^{2}} \right\}.}}}}} & \lbrack 5\rbrack\end{matrix}$

In the published work of the SEPG fitting algorithm, the amplitudeimages were obtained from fully or 60% sampled Cartesian k-space. Forhighly undersampled data (<10% sampled), the fitting algorithm relies ona reconstruction algorithm which can accurately recover the decaycurves. Thus, it is intuitive to combine the SEPG model with amodel-based algorithm (11, 22):

$\begin{matrix}{I_{0},T_{2},{B_{1} = {\arg\;{\min_{I_{0},T_{2},B_{1}}{{\quad\quad}\left\{ {\sum\limits_{j = 1}^{N}\;{{{{FT}\left\{ {C_{j}^{\prime}\left( {I_{0},T_{1},T_{2},B_{1},{\alpha_{0}(z)},\ldots\mspace{14mu},{\alpha_{j}(z)}} \right)} \right\}} - K_{j}}}^{2}} \right\}}}}},} & \lbrack 6\rbrack\end{matrix}$where I₀, T₂, B₁ are the vectors of I₀, T₂, B₁ of all voxels, FT is theforward Fourier transform, K_(j) is the (undersampled) k-space dataacquired at the j^(th). Due to the large dimensionality of the problem,it is impractical to use a global minimization algorithm to solveEquation [6], hence a local minimization algorithm (such as thegradient-based minimization algorithm) can be used. However, the SEPGmodel is non-linear as a function of T₂ and B₁ and it is difficult toobtain the gradient for minimization purposes.

In the REPCOM algorithm (12), we use PCA to linearize the exponential T₂decay model, thus overcoming non-linear minimization problems. In CURLIEwe extend the PCA approach to approximate the SEPG model. The principalcomponents (PCs) are calculated for given TE points using a training setof decay curves given by the SEPG model for a certain range of T₂ and B₁values. With the PCs obtained from the training curves, the T₂ decaycurves with indirect echoes can be approximated by a weighted linearcombination of the PCs.

Let {right arrow over (v)} be a vector representing a noiseless T₂ decaycurve with indirect echoes. Let L be the number of PCs to be used in theapproximation and {right arrow over (p)}_(i){right arrow over (b)}_(i)the i^(th) PC vector. {right arrow over (v)} can be approximated by alinear combination of {right arrow over (p)}_(i):{right arrow over (v)}=Σ _(i=1) ^(L) m _(i) {right arrow over (o)}_(i),  [7]where m_(i) is the weighting of the i^(th) principal component.{circumflex over (P)}=({right arrow over (p)}₁, {right arrow over (p)}₂,. . . , {right arrow over (p)}_(L)) is the matrix consisting of thevectors of the first L PCs. M=({right arrow over (M)}₁, {right arrowover (M)}₂, . . . , {right arrow over (M)}_(L)) Let {right arrow over(M)}_(i) be the vector of m_(i) for all the voxels and M can be formedas ({right arrow over (M)}₁, {right arrow over (M)}₂, . . . , {rightarrow over (M)}_(L)). Let {circumflex over (B)}_(j){circumflex over(P)}_(j) denote the j^(th) row of the matrix {circumflex over (P)}. Notethat M{circumflex over (B)}_(j) ^(T)M{circumflex over (P)}_(j) ^(T)yields the image at TE_(j) from the L principal component coefficients.Equation [6] can be reformulated as:{circumflex over (M)}=arg min_(M){Σ_(j=1) ^(L) ∥FT _(j)(M{circumflexover (P)} _(j) ^(T))−{right arrow over (K)} _(j)∥²}.  [8]When the complex coil sensitivity profiles S_(i) and the sparsifyingpenalties Penalty_(i)(·) (with corresponding weights λ_(i)) are given(12), the algorithm can be written as:{circumflex over (M)}=arg min_(M){Σ_(l=1) ^(#coils)Σ_(j=1) ^(L) ∥FT_(j)(S _(l) M{circumflex over (P)} _(j) ^(T))−{right arrow over (K)}_(l,j)∥²+Σ_(i)λ_(i)Penalty_(i)(M)}.  [9]The penalty terms are used to exploit the spatial compressibility of thePC coefficient maps in the framework of compressed sensing and have beenshown to improve the quality of the reconstructed T₂ maps (12). Thedecay curves are reconstructed from {circumflex over (M)} which can beobtained by a conjugate gradient minimization algorithm.

The SEPG fitting algorithm (Equation [5]) can then be applied to thecurves reconstructed by CURLIE to obtain T₂ estimates. As mentionedabove, Lebel and Wilman proposed to fix the T₁ to be +∞ for the fitting(17). As an alternative, an optimized T₁ value can be used in the SEPGfitting based on the prior knowledge of the anatomy being imaged asindicated by Sénégas et al. (23).

FIG. 1 is a functional block diagram illustrating an exemplaryembodiment of a system 100 used to perform the image processingdescribed herein. A magnetic resonance (MR) scanner 102 is used toacquire the imaging data. The MR scanner 102 is coupled to an interface104. The interface 104 provides instructions and control information tothe MR scanner 102 and receives imaging data from the MR scanner.

The system 100 also includes a central processing unit (CRU), which maybe a conventional micro-processor, a customized micro-processor, digitalsignal processor, or the like. The system is not limited by the specificform of implementation of the CPU 106.

The system 100 also includes a memory 108, which may include randomaccess memory, read-only memory, flash memory, and the like. A portionof the memory 108 may be implemented integrally with the CPU 106. Ingeneral, the memory 108 stores data and provides instructions to the CPU106 for processing the stored data.

The system 110 also includes data storage, which may be implemented as aportion of the memory 108 or a stand-alone data storage, such as amagnetic disk drive, tape drive, optical storage device, or the like. Aswill be described in greater detail below, the data storage 110 maystore the imaging data received from the MR scanner 102 via theinterface 104.

The system 100 also includes a parameter storage 112. The parameterstorage 112 may include data, such as the spin-echo pulse sequences. Theparameter storage 112 may also store modeling data for reconstruction.

The system 100 also includes an image processor 114 that will processthe image data in accordance with the processing techniques describedherein. Those skilled in the art will appreciate that the imageprocessor 114 may be a stand-alone processor or part of the CPU 106executing instructions from the memory 108 to perform the functions ofthe image processor 114. The image processor 114 is illustrated in thefunctional block diagram of FIG. 1 as a separate component because itperforms a separate function.

The system 100 also includes an image display device, which may be acomputer monitor or other conventional output device, such as a printer.

Various components of the system 100 are coupled together by a bussystem 118. The bus system 118 may include an address bus, data bus,power bus, control bus, and the like. For the sake of clarity, thosevarious busses are illustrated in FIG. 1 as the bus 118.

The flow chart of the CURLIE reconstruction algorithm and the SEPGfitting is shown in FIG. 2. Different from REPCOM, the new methodincludes the effects of indirect echoes in the T₂ estimation process byincorporating the SEPG model in the generation of the training curvesand in the estimation of the principal components coefficients (via theCURLIE algorithm) as well as in the fitting of the TE images to obtainthe final T₂ map.

Methods

All radial MESE data were acquired on a 1.5 T Signa HDxt GE (GeneralElectric Healthcare, Milwaukee, Wis.) MR scanner 102 using a previouslydeveloped radial fast spin-echo (radFSE) pulse sequence (24). A“bit-reversed” angular ordering was used to minimize artifacts from T₂decay and motion (25). The slice profiles of the excitation andrefocusing pulses were generated from the Fourier transform of the radiofrequency waveforms provided by the manufacturer for fast-spin echopulse sequences. The refocusing slice radio frequency profiles arecalibrated by the techniques in Ref. 26. It should be noted that in themanufacturer's pulse sequence the full-width half-maximum of therefocusing pulses is designed to be about 1.6 times as thick as thecorresponding excitation pulse.

Computer Simulation

Curves were generated according to the SEPG model assuming an ETL=16 andecho spacing=12.11 ms with the slice discretized into 63 points alongthe slice profile. A total of 765 curves were generated for T₂ valuesvarying from 50 ms to 300 ms with a step size of 5 ms, B₁ values varyingfrom 0.5 to 1.2 with a step size of 0.05, and T₁=+∞. The PCs weretrained from these 765 curves.

Phantom Data

A physical phantom, containing three 10 mm glass tubes filled with MnCl₂solutions of different concentrations (50 μM, 75 μM, and 170 μM) toyield data at different T₂'s, was prepared. RadFSE data were acquiredwith a single channel transmit/receive coil, with an echo train length(ETL) of 16 with echo points spaced by 12.11 ms, TR=1 s, excitationslice thickness=8 mm, receiver bandwidth=±15.63 kHz, field of view(FOV)=10 cm. The acquisition matrix for undersampled data was 256×256yielding 16 radial k-space lines for each of the 16 TE data sets. Forcomparison purposes, another data set was acquired with an acquisitionmatrix of 256×4096 to yield 256 radial k-space lines for each TE; allother acquisition parameters were kept the same. Although for radialacquisition a data set with 256 radial lines and 256 sampling points perline is 64% sampled (according to the Nyquist theorem a fully sampleddata set requires 402 radial lines for 256 sampling points per line),the data set gives good quality images and accurate T₂ maps for mostclinical applications (as well as for our phantom work). In this work,this data set is referred to as “standard sampled”.

Cartesian single-echo spin-echo data (not contaminated by indirectechoes) of the phantom were also acquired to obtain the gold standard T₂values. Data were acquired with a single channel transmit/receive coil,TR=5 s, excitation slice thickness=8 mm with the refocusing slice beingapproximately 5 times as thick as the excitation slice, receiverbandwidth=±15.63 kHz, and FOV=10 cm. Data for 4 TE points (12, 24, 36and 48 ms) were acquired with the acquisition matrix per TE point set to64×64. A lower resolution was used for the single-echo spin-echosequence due to its long acquisition time. Gold standard T₂ values wereobtained by fitting the data to a single exponential decay.

In Vivo Data

Each in vivo data set (brain, liver and heart) were acquired from aseparate volunteer under informed consent with a protocol approved bythe local Institutional Review Board.

In vivo brain data were acquired using the radFSE pulse sequence with an8-channel receiver head coil, 16 TE points equispaced by 12.93 ms,excitation slice thickness=8 mm, receiver bandwidth=±15.63 kHz, TR=4 s,and FOV=24 cm. The acquisition matrix for the standard sampled data setwas 256×4096 to yield 256 radial lines per TE. The acquisition matrixfor the undersampled data set was 256×512 yielding 32 radial lines perTE (8.0% sampled compared to fully sampled data). Experiments wereconducted with the refocusing radio frequency pulses FAs prescribed tobe 180° and 120°.

In vivo liver data were acquired in a breath hold using the radFSE pulsesequence with an 8-channel receiver torso coil, 16 TE points equispacedby 8.93 ms, slice thickness=8 mm, receiver bandwidth=±31.25 kHz, TR=1.5s, FOV=48 cm, and acquisition matrix=256×256 to yield 16 radial linesper TE (4.0% sampled compared to fully sampled data). To evaluate theestimated T₂ values, three tubes containing either MnCl₂ or agarose forT₂ variation were placed on the subject's chest. The gold standard T₂values of the three tubes were estimated to be 56.0 ms, 77.9 ms and 64.0ms in a separate experiment using a single-echo spin-echo sequence.

In vivo cardiac data were acquired using a radFSE pulse sequence whichincluded a double-inversion preparation period for nulling the signalfrom flowing blood (27). Data were acquired with an 8-channel receivercardiac coil, 16 TE points equispaced by 9.46 ms (for refocusing pulseswith prescribed FA=180) or 6.85 ms (for refocusing pulses withprescribed FA=155°), slice thickness=8 mm, receiver bandwidth=±31.25kHz, TR=1 RR, and FOV=48 cm. The acquisition matrix was 256×256resulting in 16 radial lines per TE (4.0% sampled compared to fullysampled data). Imaging of a slice was completed within a breath hold(˜15-18 s).

Data Reconstruction

All algorithms were implemented in Matlab (MathWorks, Natick, Mass.).The training curves for PC ({circumflex over (P)}) generation wereprovided by the SEPG signal model using Equation [3]. Equation [8] wassolved iteratively by using the nonlinear Polak-Ribiere ConjugateGradient algorithm (28). The spatial penalty terms used in Equation [9]consisted of the 1-norms of the wavelet transform (Daubechies 4, codeobtained from http://www-stat.stanford.edu/˜wavelab) and total variationof the PC coefficient maps.

T₂ estimates were obtained either by conventional exponential fitting orby SEPG fitting. SEPG fitting was performed using Equation [4]; duringthe fitting, T₂ was allowed to vary between 30-5000 ms and B₁ waslimited to 0-3. The T₁ values used in the SEPG fitting were fixed to anoptimal value using prior information. For each in vivo image a singleT₁ was chosen based on reported values for the anatomy of interest: graymatter for brain (950 ms) (29), liver (500 ms) (30) and myocardium (700ms) (31). For the phantom data, where there was a wide range of T₁values among the different vials used, the optimal T₁ was determined asproposed in Ref. 23. In brief, a set of simulated curves were generatedusing the SEPG model with T₁, T₂, and B₁ varying independently between300-2000 ms, 50-300 ms, and 0.5-1.2, respectively. SEPG fitting was thenperformed on these curves for each of the T₁ values in the selectedrange. The T₁ value that minimized the T₂ estimation error according tothe 2-norm was selected to be the optimized T₁.

For comparison, T₂ maps were also reconstructed using the REPCOMalgorithm. The T₂ values for the PC training were between 35 ms to 300ms equispaced by 1 ms. Total variation and wavelet transforms were usedas the sparsifying transform with proper weightings as in Ref. 12.

Results

Proof-of-Concept

In order to demonstrate that a small number of PCs are sufficient tocharacterize the T₂ decay curves in the presence of indirect echoes, wefirst performed simulations. FIG. 3A shows four out of the 765 curvesgenerated according to the SEPG model. FIG. 3A illustrates selectedtraining curves for T₂=100 ms, T₁=+0, and B₁ ranging from 0.6 to 1.2.The curves were simulated numerically using the SEPG model. An echospacing of 12.1 ms was used in the simulations to cover a TE rangesimilar to in vivo data. The curves show the expected signal modulationdue to indirect echoes with modulations increasing with B₁. FIG. 3Bshows the 2-norm of the approximation error when six PCs are used torepresent the curves after these are normalized so that the 2-norm isunity. Note that the 2-norms of the approximation error are all below0.006 and most of the errors have 2-norm <0.003. FIG. 3C shows thedistribution of the % error of the T₂ values estimated by SEPG fittingcaused by the PC approximation when six PCs are used to represent thetraining decay curves. As shown, the % errors in T₂ values estimatedfrom the curves recovered from six PCs are all <3.5% with majority ofthe errors being <1.5%. These experiments show that it is sufficient touse L=6 to represent the training curves accurately under the givenconditions and that the approximation error caused by using only thefirst six PCs is small. From our experience L depended on the samplebeing imaged. In some cases using L=3 gave good results.

Experiments using physical phantom data were performed to furtherinvestigate the accuracy of the linear approximation using principalcomponent decomposition. Phantom data were acquired with radFSE usingstandard sampled data (256 radial lines per TE with 256 points perradial line). FIGS. 4A-4D show decay curves reconstructed directly fromthe acquired data for each of the three phantom tubes shown in the imageand for various refocusing FA. Each point in the decay curvescorresponds to the averaged signal from voxels within a region ofinterest (ROI) for each tube. Each curve is the fitted decay curveaccording to the SEPG model from fitting parameters using the SEPGfitting (T₁=+∞). The curves are normalized (2-norm=1). As shown, thefitted curves are well matched to the acquired data for all the fourFAs. FIGS. 4E-4G show the approximation error of the correspondingcurves recovered from the six PCs used to generate FIGS. 3B-3C. Theerrors are all below 0.01 with most under 0.005. The 2-norms of theapproximation error are between 0.0018 and 0.0042 for all FAs which alsoshow that six PCs can represent the decay curves of the acquired datawell. Note that the same PCs were used for all FAs (FA changes areequivalent to B₁ changes under the model). In this experiment T₁ was notoptimized and assumed to be +∞.

The accuracy of T₂ estimation is then evaluated using the curvesrecovered from six PCs using the standard sampled phantom data. The datain Table 1 of FIG. 5 compares the bias in T₂ estimates obtained from the“recovered curves” to those from curves obtained directly from the TEimages (“original curves”). The data in the table of FIG. 5 correspondsto the vials in the physical phantom shown in FIG. 4 acquired withstandard radial sampling (i.e., data for each TE time point as 256radial lines with 256 points per line.) The table shows results forvarious FA and for three fitting algorithms: exponential fitting, SEPGfitting with T₁=+∞, and SEPG fitting with T₁ optimized. The percentageerrors are relative to the gold standard. It can be observed that foreach fitting algorithm the bias of T₂ estimates obtained from theoriginal and recovered curves are very similar. The differences in T₂bias are all below 0.3%, which further demonstrates the accuracy of thePC approximation for representing the acquired decay curves. The resultsalso show that the exponential fitting of MESE data overestimates T₂values. The estimation bias increases as T₂ increases and as the FAdiverges from 180° due to the more pronounced indirect echo effect; forFA=120°, the bias of the exponential fitting can be up to 18.4% (withthe larger bias for T₂=210 ms). The bias of the SEPG fitting with T₁=+∞is less than 6% which is significantly smaller than the bias of theexponential fitting. When an optimized T₁ (500 ms) was used, the largesterror is further reduced to 3.6% which demonstrates that the optimizedT₁ can be used in the SEPG fitting to improve T₂ estimation.

T₂ Estimation from Highly Undersampled Data

So far, we have demonstrated that a few principal components canaccurately represent the T₂ decay curves contaminated with indirectechoes using standard sampled data. For highly undersampled data we needto use the CURLIE algorithm described in Equation [8], where thePCA-based signal model is used to match the acquired k-space data. Usingthe three-tube physical phantom and highly undersampled radFSE data(i.e., data with only 16 radial lines per TE) we tested the accuracy ofCURLIE for reconstructing the decay curves using six PCs. FIG. 6illustrates the difference between curves reconstructed from highlyundersampled data using CURLIE and curves obtained from the standardsampled data (64% sampled) for various FAs. The highly undersampled datawas 4% relative to the Nyquist criteria. In general, the differencesbetween the curves are small despite the fact that the highlyundersampled data has 16 times less samples than the standard sampleddata. The 2-norms of the differences are between 0.0035 and 0.0095 whichdemonstrates that decay curves from highly undersampled data can beaccurately reconstructed by CURLIE. The T₂ estimates for the three tubesin the phantom, obtained with SEPG fitting, are shown in Table 2, shownin FIG. 7. Note that the T₂ bias of undersampled data is similar to theT₂ bias obtained for standard sampled data (data from Table 1): lessthan 5% for T₁=+∞ and less than 3.6% for T₁=500 ms. This demonstratesthat the decay curves reconstructed by CURLIE from highly undersampleddata have the same T₂ characteristics as the decay curves obtained fromthe standard sampled data.

CURLIE was tested in vivo using brain data. Two experiments wereconducted: one where the prescribed FA of the refocusing pulses was180°, another where the prescribed FA was 1200. The same six PCs used inTable 2 of FIG. 7 were used for the reconstruction of the undersampledbrain data. In FIG. 8 the decay curves of the shown ROI obtained fromstandard sampled data (directly from the TE images) are compared to thecurves reconstructed from 32 radial lines per TE (8.0% sampled) usingCURLIE. Note that the curves generated from under- and standard sampleddata are very similar (the error for all echo points for each of the twoFA are below 0.01). The T₂ values obtained from the decay curves usingSEPG fitting are also shown in the figure. The differences between theT₂ values obtained from under- and standard sampled data are below 3%.These results show that the curves reconstructed from highlyundersampled in vivo data agree well with the decay curves obtained fromstandard sampled data. Also, note that the T₂ values estimated from dataacquired with 180° and 120° FAs are similar.

A voxel-wise brain T₂ map obtained using CURLIE and SEPG fitting fordata acquired with prescribed FAs of 180° is shown in FIG. 9A. FIG. 9Bshows the difference map between data acquired with prescribed FAs of180° and 120° but reconstructed with the REPCOM algorithm (i.e. wherethe training curves do not take into account indirect echoes and singleexponential fitting is used for T₂ estimation). The % difference mapsbetween T₂ maps obtained by CURLIE with SEPG fitting from data acquiredwith prescribed FAs of 180° and 120° is shown in FIGS. 6C and 6D forT₁=+∞ and T₁=950 ms, respectively. The same six PCs as in previousfigures were used for both FAs. In the % difference maps, the ventriclesare masked out due to the fact that the T₂ of cerebrospinal fluid (˜2000ms) is much larger than the white and gray matter, and the TE coverageused here is not suitable for the accurate estimation of long T₂s. Themeans of the difference map are: 8.3% for the map derived from theREPCOM reconstruction and 4.9% and 3.1% for the maps derived from CURLIEand SEPG fitting with T₁=+∞ and T₁=950 ms, respectively. The small meansin the difference maps support the concept that the CURLIEreconstruction followed by SEPG fitting significantly reduces the effectof indirect echoes in T₂ estimation in vivo. When an optimized T₁ isused in the SEPG fitting the % difference T₂ map obtained by SEPGfitting with optimized T₁ is generally smaller. It is noteworthy topoint out that since the data sets with 180° and 120° FAs were obtainedby two separate acquisitions, there is slight difference between the twoimaging slices due to inter-scan movement even though the same slice wasprescribed.

An investigation of the utility of T₂ mapping with CURLIE and SEPGfitting was conducted for abdominal imaging where high undersampling isneeded due to the acquisition time constraint imposed by the breathhold. FIG. 10 shows the anatomical images and the voxel-wise T₂ maps ofan abdominal slice. The maps were reconstructed from highly undersampleddata (16 radial k-space lines per TE; 4.0% sampled with respect to theNyquist criteria) by REPCOM and CURLIE with optimized T₁ SEPG fitting.Three phantoms with known T₂ values were placed above the subject duringdata acquisition and used as the gold standard (due to the breath holdlimitation, it was not possible to obtain a gold standard data set forthe liver T₂ map). The CURLIE reconstructions were performed with L=6.The PCs were obtained from training curves generated for a T₂ range of35-350 ms, T₁=+∞ and B₁=0.5-1.2. As shown in the figure, the T₂estimates obtained from the REPCOM reconstruction increase when the FAdecreases from 180° to 120°, while the T₂ maps reconstructed by CURLIEwith SEPG fitting (optimized T₁=500 ms) are comparable for the twodifferent FAs. Note that since the data sets with 180° and 120° FAs wereobtained from two separate breath hold acquisitions the slices aresimilar but not exactly the same.

The T₂ estimates of the phantoms imaged with the liver subject aresummarized in Table 3, illustrated in FIG. 11. For both 1800 and 1200prescribed FAs, the T₂ estimates by CURLIE with SEPG fitting have <3.5%error compared to the gold standard T₂. When the effect of indirectechoes is not taken into account in the reconstruction (as in REPCOM),the error can be up to 14% and 38% for FAs of 180° and 120°,respectively. The trend is similar as that shown in FIG. 10, and thefact that the T₂ values obtained from CURLIE with SEPG fitting are verysimilar to the gold standard T₂ values indicates that the CURLIE T₂ mapsshown in FIG. 10 are accurate.

CURLIE with optimized T₁ SEPG fitting can also be applied for T₂ mappingof the myocardium using data acquired in a single breath hold. Incardiac imaging it is desirable to keep the echo train as short aspossible to avoid cardiac motion during data acquisition thus, a shortrefocusing pulse (i.e., FA<1800) is typically used. FIG. 12 shows theanatomical short-axis cut of the heart acquired with a double-inversionradFSE sequence and T₂ maps reconstructed via CURLIE with SEPG fitting(optimized T₁=700 ms) and REPCOM. The maps were reconstructed from 16radial k-space lines per TE (4.0% sampled). Similar to the observationmade for the liver, the T₂ maps reconstructed by CURLIE with SEPGfitting are comparable for the two different refocusing pulses, whereasthe T₂ maps reconstructed by REPCOM show a greater disagreement betweenthe two refocusing pulses.

Discussion

The CURLIE process has been shown to accurately reconstruct the decaycurves with indirect echoes from highly undersampled radial MESE data.CURLIE uses a linear approximation of the signal decay allowing for theincorporation of the highly non-linear SEPG model to account for theeffects of indirect echoes. Moreover, the TE images generated via CURLIEhave high spatial and temporal resolution. As a result, CURLIE combinedwith SEPG fitting enables accurate T₂ estimation from highlyundersampled radial MESE data allowing for the reconstruction of T₂ mapsfrom data acquired in a short period of time. For instance T₂ maps ofthe whole brain can be achieved in 4 min or less (depending on thedegree of undersampling used). Maps of the thoracic cavity and abdomencan be obtained in a breath hold.

As shown in this work, the indirect echoes which are inherent to MESEacquisitions cause a significant (positive) T₂ bias if data arereconstructed assuming an exponential decay. The indirect echo effect ismore pronounced for longer T₂s and as the FA of the refocusing pulsesdeviate from the ideal 180° (17). Thus, without indirect echocompensation, the T₂ estimates from MESE data will depend on the profileof the radio frequency pulse, B₁ imperfections, as well as the TEcoverage used in the experiment (number of TE points and echo spacing).As a result, the inter-site or inter-scan reproducibility of T₂measurements can be greatly impacted. Indirect echoes also limit the useof MESE for T₂ mapping at higher fields (field strength ≥3T) where SARlimits the use of 180° refocusing pulses or in cardiac applicationswhere shorter refocusing pulses are used to reduce the acquisitionwindow and minimize the effects of motion.

Our results showed that when curves are reconstructed with CURLIEfollowed by SEPG fitting the T₂ bias of phantoms (compared to a goldstandard) are small and not dependent on the T₂ values and FAs of therefocusing pulses even for data acquired with a high degree ofundersampling. The same trend in seen vivo: the T₂ maps of brain, liverand heart reconstructed from highly undersampled data with CURLIE andSEPG fitting are not affected by the prescribed FA of the refocusingpulses as those reconstructed from the REPCOM algorithm. Overall, thetechnique should provide a fast method for T₂ mapping that is lessdependent on the experimental conditions including the magnetic fieldsstrength. These unique characteristics should make the techniquepractical for clinical use.

In this work SEPG fitting was performed for T₁=+∞ or an optimized T₁,however, in the reconstruction of the decay curves via CURLIE, T₁ wasfixed to infinity. This can be optimized using prior information of theobject being imaged and the scanner and imaging parameters. Similaroptimization can be performed for the T₂ and B₁ for the generation ofthe PCs. The T₂ range used to generate the training curves can also beoptimized by using prior knowledge based on the anatomy being imaged, orestimated by prior T₂ mapping using a different algorithm (e.g. REPCOMor the echo sharing algorithm described in Ref. 14). In this work weused 6 principal components to approximate the signal model and showedthat the T₂ can be accurately estimated in phantoms and a series of invivo applications such as brain, myocardium and liver. In this work, thenumber of principal components (L) was determined empirically for thegiven training set. It is expected that smaller L could yield similarresults for an optimized training set. However, given the ranges and/ordistributions of T₂, T₁ and B₁ the optimal L and design of the trainingcurves remain open problems.

Although SEPG fitting yields fitted B₁ maps in addition to the T₂ maps,the fitted B₁ maps estimated from our algorithm showed T₁ effects. Thesemanifested as fluctuations along interfaces between tissues with verydifferent T₁s (e.g., gray-white matter tissue and CSF or liverparenchyma and blood vessels). However, the nature of the B₁ maps didnot affect the T₂ results. We verified this experimentally by comparingthe T₂ maps obtained from the CURLIE-SEPG method to those resulting fromjust SEPG fitting (using standard sampled data when available) or by ausing smoothing constraint for the B₁ maps before T₂ estimation. The T₂maps were similar regardless of the B₁ maps.

For the non-optimized Matlab code used here, the reconstruction tookabout 40 min using a single core of a desktop computer (Intel Core 2Quad CPU, 2.4 GHz) for a single slice when the data were acquired witheight coils, 256 k-space lines. However, a significant reduction inreconstruction time is expected when the reconstruction code isoptimized and parallelized since most of the computation time was spenton matrix multiplication.

Joint Bi-Exponential Fitting Processing

In another aspect, the linearization approach can be used to estimate T₂from highly undersampled data that obeys a signal model where multipletissue components or chemical species are contained in a voxel. In thesecases the measurements obtained are a linear combination from more thanone component (with different weightings) and noise. Thus, thelinearization process can deal better with signal models based onmultiple components. For example, if PCs are used as a linearizationtool, a multiple component system can be represented by a linearcombination of the PCs. However, a problem encountered with fittingalgorithms that handle multi-component data is that the parameterestimation is significantly more affected by the level of noise in themeasurement data compared to single component systems. For a multiplecomponent model the techniques described herein utilize linearcombination weighting variation to improve accuracy of estimation byfitting the data jointly (i.e., joint bi-exponential fitting processingor JBF). One application of the joint approach is to correct for theeffects of partial volume in quantitative magnetic resonance imagingacquired from highly undersampled data.

For the parameter estimation involving multi-component systemsconventional approaches increase the amount of measurements in order toincrease the number of points needed to fit the data and/or increase thesignal to noise ratio of each data point. As discussed above, theseapproaches typically require long acquisition times which in many cases(such as in clinical MRI) is not a viable option. As an alternative toincreasing the amount of measurements, others have proposed to averageinformation within a region-of-interest (ROI); this is referred to asthe region fitting (RF) algorithm. The RF algorithm does not take intoaccount the natural variations occurring within the ROI. For example, inthe case of parameter estimation for small liver lesions, there isvariation in the lesion fraction (relative amount of lesion and livertissues) within a given imaging voxel. This natural variation is ignoredin the conventional RF approach. The joint fitting approach describedherein utilizes data representative of this variation to improveparameter estimation.

As will be described in greater detail below, the JBF processingprovides fast and accurate parameter estimation in the presence ofpartial volume from MRI data acquired in a short period of time. Thecommercial impact could be considerable given that the concept could beapplied across vendor platforms and across an array of acquisitiontechniques as well as disease types. Specific technique application forliver lesion imaging is provided in this disclosure. Liver imaging isone of the largest growing applications for clinical MRI and the JBFprocessing could have significance for applications including tumorcharacterization. Those skilled in the art will appreciate that the JBFtechniques described herein can be applied to other tissue types besidethe examples of liver tissue described herein.

As previously discussed, MRI techniques have a significant limitation ofacquisition time. For example, experimental data is shown where the JBFprocessing combined with a radial MRI technique is used to yield imageswith high spatial and temporal resolution from which accurate T₂ maps oftissues are obtained in the presence of partial volume effects. Normallythis is not feasible with conventional techniques within importanttissues of interest, such as the liver, where data needs to be acquiredin a short period of time (e.g., breath hold) to mitigate the effects ofmotion, as related to breathing. The techniques described herein allowthe imaging data to be obtained within a breath hold period. While thedata collected within a breath hold is highly undersampled, the JBFprocessing can still generate images with quality that matches imagequality of longer data acquisition times. Currently, no other techniquesare available to achieve this degree of acceleration when high temporaland spatial resolution is required for tissue characterization.

Parameter estimation is widely used in many fields. In many situations,the measurements obtained are a linear combination from more than onecomponent (with different weightings) and noise. For parameterestimation from multi-component data in the presence of noise we proposea novel technique which uses a joint bi-exponential fitting (JBF)approach. The new algorithm yields accurate estimates with significantlyless data compared to conventional fitting methods.

The JBF algorithm is described here in the context of T₂ estimation forsmall structures (e.g., tumors or lesions) embedded in a background(e.g., tissue parenchyma such as liver, brain, or muscle). However, themethodology can be extended to other applications in the field ofmedical imaging or imaging technology in general.

T₂ Estimation for Small Tumors Embedded in a Background Tissue

The characterization of small lesions embedded in a tissue backgroundcan be challenging due to the contamination from the background. This isknown as the partial volume effect or PVE and is the main source oferror in small lesion classification based on techniques such asT₂-weighted, diffusion-weighted MRI, and contrast enhancement techniques(32-34).

If we consider a lesion with a diameter <15 mm and a 6-8-mm thickimaging slice, most of the voxels within the ROI containing the lesionare contaminated with signal from the background tissue. Thus, anyparameter estimated for the lesion will be affected by the PVE. In thecase of T₂ estimation, the PVE can be accounted for by using thefollowing multi-exponential signal model:s(t)=I _(i) e ^(−t/T) ^(2l) +I _(b) e ^(−t/T) ^(2b) +ε(t),  [10]where I_(l), I_(b) are the initial signal intensities of the lesion andthe background tissue (i.e., signal at time t=0 ms), T_(2l), T_(2b) arethe corresponding T₂ values, and ε(t) is the noise of the given voxel attime t. Given the signals of a voxel at N TE time points, least-squarefitting can be used to estimate the parameters I_(l), I_(b), T_(2l),T_(2b) in a voxel-wise fashion:

$\begin{matrix}{{\hat{I}}_{l},{\hat{I}}_{b},{\hat{T}}_{2l},{{\hat{T}}_{2b} = {\underset{I_{l},I_{b},T_{2l},T_{2b}}{argmin}{\left\{ {\sum\limits_{n = 1}^{N}\;{{{I_{l}e^{{- {TE}_{n}}/T_{2l}}} + {I_{b}e^{{- {TE}_{n}}/T_{2b}}} - {s\left( {TE}_{n} \right)}}}^{2}} \right\}.}}}} & \lbrack 11\rbrack\end{matrix}$

Due to the nature of the multi-exponential model and the presence ofnoise, the conventional voxel-wise bi-exponential fitting (VBF)algorithm leads to large uncertainty of the fitted parameters (35). Oneway to improve fitting for bi-exponential models is to reduce the noise.Thus, as an alternative to VBF, the RF approach was proposed (36). Thealgorithm is based on averaging the data from all the voxels within theROI at each time point prior to fitting with Equation [11]. RF is morestable than VBF (35), however, the averaging procedure disregards theinformation on the lesion fraction

${LF} = \frac{I_{l}}{I_{l} + I_{b}}$from each voxel.The Joint Bi-Exponential Fitting (JBF) Algorithm (37):

For the case of small lesions, the LF may vary significantly from voxelto voxel and this variation can be utilized to improve fitting in abi-exponential model. In order to utilize this variation we propose a T₂estimation algorithm that estimates T_(2l), T_(2b), for all voxelsjointly. The joint T₂ estimation relies on the assumption that for smalllesions (diameter <15 mm) the T_(2l), T_(2b), for all voxels within thelesion's ROI can be considered homogenous which is a realisticassumption for small lesions.

Let I_(l) ¹, I_(l) ², . . . , I_(l) ^(M) and I_(b) ¹, I_(b) ², . . . ,I_(b) ^(M) be the initial signal intensities of lesion and background,respectively, for each of the M voxels within the lesion's ROI. Lets¹(TE_(n)), s²(TE_(n)), . . . , s^(M) (TE_(n)) be the signals from theseM voxels at time TE_(n). Under the assumption of homogeneity, we canconstrain T_(2l) ^(m), T_(2b) ^(m) to two global values T _(2l) and T_(2b) within the lesion's ROI. The JBF process for T₂ estimation can beformulated as:

$\begin{matrix}{{\hat{\overset{\_}{T}}}_{2l},{{\hat{\overset{\_}{T}}}_{2b} = {\arg\mspace{14mu}\min{\left\{ {\sum\limits_{n = 1}^{N}\;{\sum\limits_{m = 1}^{M}\;{{{I_{l}^{m}e^{{- t_{n}}/{\overset{\_}{T}}_{2l}}} + {I_{b}^{m}e^{{{- t_{n}}/{\overset{\_}{T}}_{2b}}\;}} - {s_{m}\left( {TE}_{n} \right)}}}^{2}}} \right\}.}}}} & \lbrack 13\rbrack\end{matrix}$JBF Versus the Conventional VBF and Region Fitting Algorithms:

Simulations were conducted to compare the VBF, region fitting and JBFalgorithms for the specific task of estimating T_(2l) for small lesions.FIG. 13 shows plots of the estimated T_(2l) versus lesion diameteraccording to the VBF, RF, and JBF algorithms. Simulations are presentedfor mean T_(2l) values of 180 ms and 100 ms with a mean T_(2b) of 40 ms.These represent common values observed in focal liver lesions and liverparenchyma. In FIG. 13, the errors are plots of estimations by differentalgorithms versus the lesion diameter are shown. Data points that haveout-of-range error bars are not shown.

As seen in the FIG. 13, the conventional VBF approach suffers from largebiases; the bias in the estimated T_(2l) can be up to ±35%. The RFapproach suffers from large uncertainties. In fact, some of the errorbars in FIG. 13 for the RF algorithm were well out of the range of theplots thus, the data are not shown. The proposed JBF algorithm hassignificantly smaller standard deviations and more accurate meanscompared to the other two algorithms, especially for lesions withdiameters smaller than 10 mm.

In FIG. 13 we used an ROI consisting of voxels containing the lesion(tight ROI). In practice, a tight ROI may not always be achievable,particularly for small lesions. Under the assumption that the voxelsnext to the lesion's ROI have similar T_(2b) as the background voxelswithin the ROI, it should be possible to obtain accurate T_(2l)estimates even if the ROI is not tight. To prove this concept anexpanded ROI was generated by including all immediate neighboring voxelsto the tight ROI. FIG. 14 shows the performances of the three algorithmsfor the tight and expanded ROIs. The VBF estimates varied greatly whenthe expanded ROI was used. The error bars of RF have large overlapbetween the two ROIs, but the variations of the bias between the twoROIs for small diameters are large than that of JBF. For JBF thedifference of the means obtained by the two ROIs is <5%. Thisdemonstrates the insensitivity to ROI drawing of the JBF algorithm.

A study on a physical phantom was also conducted. A phantom composed offour glass tubes ending in a spherical bulb was used to represent smallspherical lesions. The tubes were filled with Magnevist (BayerHealthCare Pharmaceuticals Inc., Germany) solutions of differentconcentrations to yield different T₂ values. The tubes were inserted ina background solution that corresponded to T_(2b)˜40 ms.

The physical phantom results are summarized Table 4, shown in FIG. 15.Table 4 illustrates the results from the physical phantom showing thepercent error in T₂ estimates and compares the % errors of the estimatedT_(2l) values obtained by the VBF, RF, and JBF algorithms with respectto the gold standard (obtained from a separate single-echo spin-echoexperiment using samples with no background) for a tight and expandedROI. The results show that VBF has the largest biases and the method isvery sensitive to ROI drawing; the % error can vary up to almost 80%between a tight and an expanded ROI. RF has smaller biases than VBF andthe method is less dependent on the ROI drawing; the variation caused bythe two ROIs is under 35%. The proposed JBF algorithm is the mostaccurate (bias <3%) and the method is highly insensitive to ROI drawing;the largest observed variation for different ROIs is 1%.

Extension of JBF for Undersampled Radial FSE Data (38):

Because in clinical MRI the acquisition of data for parameter mappingneeds to be within the time constraints of a clinical examination (inmany cases a breath hold time), the amount of data available forparameter estimation is limited.

For the case of T₂ mapping the goal is to use the JBF approach withundersampled FSE data. In a breath hold experiment this means that wewill have 1/16 of the data typically required for T₂ mapping. Thus, wepropose combining JBF with a linearized-model-based reconstruction,recently developed by our group for T₂ mapping using undersampled radFSEdata (12). We named the combined methodology PURIFY for Partial volUmecoRrected rol-based T₂ Fitting of highly undersampled data.

We tested PURIFY using the physical phantom with Magnevist solutions toyield T_(2l) of 69.0 ms and 179.8 ms and T_(2b)=40.1 ms. In theexperiment we used undersampled radFSE data acquired in only 22 s. TheT_(2l) estimates obtained from JBF and single exponential fitting (i.e.,no partial volume correction) are shown in Table 5, illustrated in FIG.16. The data in Table 2 results from a physical phantom showing T₂estimates for the JBF using highly undersampled data. Note that withsingle exponential fitting the T₂ values are significantly lower thanthe gold standard due to contamination from background tissue. When JBFis used the T_(2l) values are close to the gold standard. This provesthe concept that T₂ mapping with partial volume correction using highlyundersampled data is achievable with the JBF approach.

PURIFY was also tested in vivo. FIG. 17 shows the image corresponding toan abdominal scan of a subject with a small hemangioma. The figure alsoshows a plot of lesion classification based on T₂ values generated froma set of 41 patients with larger liver lesions (>1.5 cm in diameter)(12, 24). Note that when a single exponential fit is used, the T₂ of thehemangioma falls within the range of malignant lesions (false positive)because the PVE due to liver contamination is not taken into account.When the JBF algorithm is used, the lesion is within the range of benignlesions (true negative). It is noteworthy to mention that the lesionclassification was possible from data acquired in a breath hold (22 s).

CONCLUSIONS

In this work, it has been shown by numerical simulation, phantom and invivo data that the proposed CURLIE algorithm can accurately reconstructthe decay curves with indirect echoes from highly undersampled radialMESE data. Accurate T₂ estimates can then be derived from the TE curvesvia SEPG fitting. The use of highly undersampled radial MESE data allowsfor the fast acquisition of data. The correction for indirect echoesreduces inaccuracies in T₂ mapping due to imperfect refocusing pulsesmaking T₂ mapping with MESE accurate when the FA of the refocusingpulses are less than the ideal 180°; this is particularly important athigher magnetic fields as well as in certain cardiac applications.Overall, the CURLIE algorithm combined with SEPG fitting enables fast T₂estimation which is less dependent on the experimental conditions usedfor data acquisition.

In another example of image processing using highly undersampled imagingdata, the partial volume correction process of PURIFY provides veryaccurate results with a minimal amount of data. These improvementsshould make T₂ mapping more practical for clinical use.

The foregoing described embodiments depict different componentscontained within, or connected with, different other components. It isto be understood that such depicted architectures are merely exemplary,and that in fact many other architectures can be implemented whichachieve the same functionality. In a conceptual sense, any arrangementof components to achieve the same functionality is effectively“associated” such that the desired functionality is achieved. Hence, anytwo components herein combined to achieve a particular functionality canbe seen as “associated with” each other such that the desiredfunctionality is achieved, irrespective of architectures or intermedialcomponents. Likewise, any two components so associated can also beviewed as being “operably connected”, or “operably coupled”, to eachother to achieve the desired functionality.

While particular embodiments of the present invention have been shownand described, it will be obvious to those skilled in the art that,based upon the teachings herein, changes and modifications may be madewithout departing from this invention and its broader aspects and,therefore, the appended claims are to encompass within their scope allsuch changes and modifications as are within the true spirit and scopeof this invention. Furthermore, it is to be understood that theinvention is solely defined by the appended claims. It will beunderstood by those within the art that, in general, terms used herein,and especially in the appended claims (e.g., bodies of the appendedclaims) are generally intended as “open” terms (e.g., the term“including” should be interpreted as “including but not limited to,” theterm “having” should be interpreted as “having at least,” the term“includes” should be interpreted as “includes but is not limited to,”etc.). It will be further understood by those within the art that if aspecific number of an introduced claim recitation is intended, such anintent will be explicitly recited in the claim, and in the absence ofsuch recitation no such intent is present. For example, as an aid tounderstanding, the following appended claims may contain usage of theintroductory phrases “at least one” and “one or more” to introduce claimrecitations. However, the use of such phrases should not be construed toimply that the introduction of a claim recitation by the indefinitearticles “a” or “an” limits any particular claim containing suchintroduced claim recitation to inventions containing only one suchrecitation, even when the same claim includes the introductory phrases“one or more” or “at least one” and indefinite articles such as “a” or“an” (e.g., “a” and/or “an” should typically be interpreted to mean “atleast one” or “one or more”); the same holds true for the use ofdefinite articles used to introduce claim recitations. SEP In addition,even if a specific number of an introduced claim recitation isexplicitly recited, those skilled in the art will recognize that suchrecitation should typically be interpreted to mean at least the recitednumber (e.g., the bare recitation of “two recitations,” without othermodifiers, typically means at least two recitations, or two or morerecitations).

Accordingly, the invention is not limited except as by the appendedclaims.

REFERENCES

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The invention claimed is:
 1. A system for processing magnetic resonanceimaging data from an MRI device, comprising: an interface configured toreceive the imaging data generated by the MRI device, the imaging databeing generated using a multi-echo spin-echo (MESE) pulse sequence andbeing highly undersampled; a set of training curves generated fromimaging data for a range of expected T₂ values; a set of principalcomponents derived from the set of training curves, the principalcomponents being representative of T₂ decay curves in the presence ofindirect echoes generated by imperfections in the radio frequencyrefocusing pulses associated with the MRI device, wherein the refocusingpulses are not 180° across the entire imaging slice/volume; and aprocessor configured to apply the set of principal components to theimaging data to compensate for the imperfections in the refocusingpulses used in the MRI device that distort the generated imaging data tothereby generate a corrected T₂ map that reduces T₂ value dependence onthe imperfection in the MRI device and to display a more accurate MRIimage.
 2. The system of claim 1 wherein the set of training curves aregenerated using a slice-resolved extended phase graph model or a signalmodel derived from the Bloch equations.
 3. The system of claim 1 whereinthe set of principal components comprises a set of three to sixprincipal components.
 4. The system of claim 1 wherein the processor isconfigured to process the image using:{circumflex over (M)}=arg min_(M){Σ_(l=1) ^(#coils)Σ_(j=1) ^(L) ∥FT_(j)(S _(j) M{circumflex over (P)} _(j) ^(T))−{right arrow over (K)}_(l,j)∥²+Σ_(i)λ_(i)Penalty_(i)(M)}. where {right arrow over (M)}_(i) isthe vector of m for all the voxels and M can be formed as ({right arrowover (M)}₁, {right arrow over (M)}₂, . . . , {right arrow over(M)}_(L)), {circumflex over (P)}=({right arrow over (p)}₁, {right arrowover (p)}₂, . . . , {right arrow over (p)}_(L)) is a matrix consistingof the vectors of the first L principal components where {circumflexover (P)}_(j) denotes the j^(th) row of the matrix {circumflex over(P)}, FT is the forward Fourier transform, K_(j) is the (undersampled)k-space data acquired at the j^(th) TE, and the complex coil sensitivityprofiles S_(l) and the sparsifying penalties Penalty_(i)(⋅) (withcorresponding weights λ_(i)) are given.
 5. A system comprising: amagnetic resonance imaging (MRI) device configured to generate imagingdata, the generated image data being generated using a multi-echospin-echo (MESE) pulse sequence and being highly undersampled; aninterface configured to receive the imaging data generated by the MRIdevice; and a processor configured to process the received imaging datausing a linear approximation to a signal model characterizing the T₂decay to thereby generate a corrected T₂ estimation map to compensatefor errors in the highly undersampled generated imaging data arisingfrom indirect echoes generated by imperfections in the radio frequencyrefocusing pulses associated with the MRI device, the refocusing pulseshaving a flip angle that deviates from 180°, or multiple components dueto the presence of different tissue species within a voxel to therebygenerate and display a more accurate MRI image.
 6. The system of claim 5wherein the generated imaging data is radial fast-spin-echo acquisitiondata.
 7. The system of claim 5 wherein the processor is configured toprocess the received imaging data to compensate for indirect echoes inthe generated image data.
 8. The system of claim 5 wherein the processoris configured to process the received imaging data to generate anestimation of the T₂ relaxation time.
 9. The system of claim 5 whereinthe processor is configured to process the received imaging data toapply a linearization model to the received image data and to compensatefor indirect echoes in the generated image data.
 10. The system of claim5, further comprising: a set of training curves generated for a range ofexpected T₂ values; and a set of coefficients derived from a linearapproximation model and the set of training curves, the coefficientsbeing representative of T₂ decay curves in the presence of indirectechoes, wherein the processor is further configured to apply the set ofcoefficients to the imaging data to compensate for imperfections in therefocusing pulses in the MRI device that distort the generated imagingdata to thereby generate the T₂ map.
 11. The system of claim 10 whereinthe coefficients are derived from a group comprising principalcomponents, a manifold, and a dictionary.
 12. The system of claim 5wherein the processor is configured to process the received imaging datafrom an imaging sample containing more than one tissue or chemicalcomponent within a voxel.
 13. The system of claim 12 wherein theprocessor is configured to process the received imaging data using a setof initial lesion intensity values and a set of initial backgroundintensity values for each voxel in a designated region of interest (ROI)containing an image of a lesion and to analyze the sets of initialintensity values in a fitting process to thereby generate the T₂ maprepresentative of the lesion.
 14. The system of claim 13 wherein theprocessor is configured to analyze the sets of initial intensity valuesusing a joint bi-exponential fitting algorithm.
 15. A method forprocessing magnetic resonance imaging (MRI) imaging data generated by anMRI device, comprising: receiving the imaging data, the imaging databeing generated using a multi-echo spin-echo (MESE) pulse sequence andbeing highly undersampled; generating a set of training curves for arange of expected T₂ values; deriving a set of coefficients from alinear approximation model and the set of training curves, thecoefficients being representative of T₂ decay curves in the presence ofindirect echoes in the imaging data caused by imperfections in the radiofrequency refocusing pulses generated by the MRI device, wherein therefocusing pulses are not 180° across the entire imaging slice/volume;and applying the set of coefficients to the imaging data to compensatefor the indirect echoes that distort the imaging data to therebygenerate a corrected T₂ map that reduces T₂ value dependence on theimperfection in the MRI device and to display a more accurate MRI image.16. The method of claim 15 wherein the coefficients are derived from agroup comprising principal components, a manifold, and a dictionary. 17.The method of claim 15 wherein generating the set of training curvescomprises generating the set of training curves using a slice-resolvedextended phase graph model.
 18. The method of claim 15 wherein the setof principal components comprises a set of three to six principalcomponents.
 19. A method for processing imaging data generated by amagnetic resonance imaging (MRI) data comprising: receiving the imagingdata, the imaging data being generated using a multi-echo spin-echo(MESE) pulse sequence and being highly undersampled; and processing thereceived imaging data using a linear approximation to the signal modelto thereby generate a corrected T₂ estimation map to compensate forerrors in the highly undersampled generated imaging data arising fromindirect echoes generated by imperfections in the radio frequencyrefocusing pulses associated with the MRI device, the refocusing pulseshaving a flip angle that is not perfectly 180° across the entire imagingslice/volume, or multiple components due to the presence of differenttissue species within a voxel to thereby generate and display a moreaccurate MRI image.
 20. The method of claim 19 wherein processing thereceived imaging data comprises generating a T₂ estimation map tothereby compensate for indirect echoes in the imaging data.
 21. Themethod of claim 19 wherein processing the received imaging datacomprises generating an estimation of the T₂ relaxation time.
 22. Themethod of claim 19 wherein processing the received imaging datacomprises applying a linearization model to the received imaging dataand compensating for indirect echoes in the generated image data. 23.The method of claim 19; further comprising: generating a set of trainingcurves for a range of expected T₂ values; deriving a set of coefficientsfrom a linear model and the set of training curves, the principalcomponents being representative of T₂ decay curves in the presence ofindirect echoes; and applying the set of coefficients to the imagingdata to compensate for refocusing pulses in the MRI device that distortthe generated imaging data to thereby generate the T₂ map.
 24. Themethod of claim 23 wherein the coefficients are derived from a groupcomprising principal components, a manifold, and a dictionary.
 25. Themethod of claim 19 wherein processing the received imaging datacomprises processing the received imaging data from an imaging samplecontaining more than one tissue or chemical component within a voxel.26. The method of claim 25 wherein processing the received imaging datauses a set of initial lesion intensity values and a set of initialbackground intensity values for each voxel in a designated region ofinterest (ROI) containing an image of a lesion and analyzing the sets ofinitial intensity values in a fitting process to thereby generate the T₂map representative of the lesion.
 27. The method of claim 26 whereinanalyzing the sets of initial intensity values uses a jointbi-exponential fitting algorithm.